To see this we can refermionize the theory Eq. (REF ) by defining \(\psi (x,t)=\sqrt{\frac{D}{2\pi }}:e^{i\sqrt{2}\phi ^o(x,t)}:f\) , with \(f^\dagger =f\)  being a Majorana zero mode (MZM) [1]}, [2]}, [3]} and \(D\)  is the bandwidth. The fermionic fields obey the equal time commutation relations: \(\lbrace \psi (x),\psi ^\dagger (x^{\prime })\rbrace =\delta (x-x^{\prime })\) , \(\lbrace f,f\rbrace =2\) , and \(\lbrace \psi (x),f\rbrace =0\) . The fermionized odd boson Hamiltonian becomes
\(H^o_{\nu =\frac{1}{2}}=-i\int _{-\frac{L}{2}}^{\frac{L}{2}} dx\psi ^\dagger (x)\partial _x \psi (x)+\sqrt{\frac{\pi }{2D}} \lambda (\psi (0)-\psi ^\dagger (0))f,\) 
